Problem: Factor the following expression: $4$ $x^2+$ $7$ $x+$ $3$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(4)}{(3)} &=& 12 \\ {a} + {b} &=& & & {7} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $12$ and add them together. The factors that add up to ${7}$ will be your ${a}$ and ${b}$ When ${a}$ is ${3}$ and ${b}$ is ${4}$ $ \begin{eqnarray} {ab} &=& ({3})({4}) &=& 12 \\ {a} + {b} &=& {3} + {4} &=& 7 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {4}x^2 +{3}x +{4}x +{3} $ Group the terms so that there is a common factor in each group: $ ({4}x^2 +{3}x) + ({4}x +{3}) $ Factor out the common factors: $ x(4x + 3) + 1(4x + 3) $ Notice how $(4x + 3)$ has become a common factor. Factor this out to find the answer. $(4x + 3)(x + 1)$